Higgs Factory options for CERN: A comparative study

``All future $e^+e^-$ Higgs factories have similar reach for the precise measurement of the Higgs boson properties.'': this popular statement has often led to the impression that all $\rm e^+e^-$ options are scientifically equivalent when it comes to choosing the future post-LHC collider at CERN. More recently, the concept of sustainability has been added in attempts to rank Higgs factories. A comparative analysis of the data currently available is performed in this note to clarify these issues for three different options: the future circular colliders (FCC), and two linear collider alternatives (CLIC and ILC@CERN). The main observation is as follows. For the precise measurement of already demonstrated Higgs decays (b\=b, $\tau^+\tau^-$, gg, ZZ, WW) and for $\rm H \to c\bar c$, it would take half a century to CLIC and ILC@CERN to reach the precisions that FCC-ee can achieve in 8 years thanks to its large luminosity and its four interactions points. The corresponding electricity consumption, cost and carbon footprint would also be very significantly larger with linear colliders than with FCC-ee. Considering in addition that (i) [...]; (ii) [...]; (iii) [...]; and {\it (iv)} the vast experimental programme achievable with both FCC-ee and FCC-hh is out of reach of linear colliders; it is found that FCC-ee is a vastly superior option for CERN, and the only first step en route to the 100\,TeV hadron collider.